So the number of ways N people can be arranged in a straight line if 2 particular people must be separated

To solve this problem, we can use the concept of permutations and combinations in probability.

Step 1: Consider the two people who must be separated as one entity. So, we now have (N-1) entities.

Step 2: These (N-1) entities can be arranged in (N-1)! ways.

Step 3: The two people who are considered as one entity can be arranged among themselves in 2! ways.

Step 4: Therefore, the total number of ways in which N people can be arranged such that the two particular people are never together is (N-1)! * 2!.

Step 5: However, this is the number of ways in which they are together. We need to find the number of ways in which they are not together.

Step 6: The total number of arrangements of N people is N!.

Step 7: So, the number of ways in which the two people are not together is the total arrangements minus the arrangements in which they are together.

Step 8: Therefore, the required number of ways = N! - (N-1)! * 2!.